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A114583 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)). 2
1, 1, 2, 3, 1, 7, 2, 15, 6, 36, 14, 1, 85, 39, 3, 209, 102, 12, 517, 280, 37, 1, 1303, 758, 123, 4, 3312, 2085, 381, 20, 8510, 5730, 1194, 76, 1, 22029, 15849, 3657, 295, 5, 57447, 43914, 11187, 1056, 30, 150709, 122090, 33903, 3734, 135, 1, 397569, 340104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 1 yields A114584. Sum(k*T(n,k),k=0..floor(n/3))=A005717(n-2).

LINKS

Alois P. Heinz, Rows n = 0..300, flattened

Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.

Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.

FORMULA

G.f.=G=G(t, z) satisfies G=1+zG+z^2*G(tz-z+G).

EXAMPLE

T(5,1)=6 because we have HH(UHD), UD(UHD), (UHD)HH, (UHD)UD, H(UHD)H and U(UHD)D, where U=(1,1),H=(1,0),D=(1,-1) (the UHD's are shown between parentheses).

Triangle begins:

   1;

   1;

   2;

   3,  1;

   7,  2;

  15,  6;

  36, 14, 1;

  ...

MAPLE

G:=(1-z-t*z^3+z^3-sqrt((1-3*z+z^3-t*z^3)*(1+z+z^3-t*z^3)))/2/z^2: Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 0 to 17 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/3)) od; # yields sequence in triangular form

# second Maple program:

b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,

     `if`(x=0, 1, b(x-1, y, `if`(t=1, 2, 0))+b(x-1, y-1, 0)*

     `if`(t=2, z, 1)+b(x-1, y+1, 1))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):

seq(T(n), n=0..15);  # Alois P. Heinz, Feb 01 2019

MATHEMATICA

CoefficientList[#, t]& /@ CoefficientList[(1 - z - t z^3 + z^3 - Sqrt[(1 - 3z + z^3 - t z^3)(1 + z + z^3 - t z^3)])/2/z^2 + O[z]^17, z] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

CROSSREFS

Cf. A001006, A114584, A115717.

Sequence in context: A114858 A193491 A263340 * A114581 A328398 A085588

Adjacent sequences:  A114580 A114581 A114582 * A114584 A114585 A114586

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 09 2005

STATUS

approved

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Last modified June 21 06:09 EDT 2021. Contains 345358 sequences. (Running on oeis4.)